How to find a group isomorphic to $D_8$ consisting of elements of $GL(2,\Bbb C)$?
How about $GL(2,\Bbb R)$?
Answer: As we know $D_8=\{e, a, a^2, a^3, b, ab, a^2b, a^3b\}$ and $a^4=b^2=e, ba= a^3b$.
Obviously: $$e=\left[\begin{matrix} 1&0\\ 0&1\\ \end{matrix}\right]$$
If $$a=\left[\begin{matrix} 0&1\\ -1&0\\ \end{matrix}\right]$$ and $$b=\left[\begin{matrix} 0&1\\ 1&0\\ \end{matrix}\right]$$ Is it true?
We should prove $\phi: D_8 \rightarrow G$ is $1-1$ and onto and $$\phi(a*b)=\phi(a)\cdot\phi(b)$$
Thank you for your answers.
The isometries of a square in $\mathbb{R}^2$ form a group isomorphic to $D_8$. If you center the square at the origin, each isometry can be represented by a linear transformation (like a rotation or a reflection), which can be written as a $2 \times 2$ real matrix.